We are now going to look at a special class of tests that give us the ability to do statistical analyses in
circumstances when parametric tests just won't do. They are called
non-parametric statistics, and I'm going to tell you why we need them.
From all of our studies of parametric statistics like t-tests and ANOVA,
we have learned how to compare groups using scale level data.
But what if you only have nominal data?
Well, you have options.
Let's imagine that we are having an end-of-semester party and we need to order soft drinks.
Somebody suggests that we just buy a case of whatever is on sale because people will drink
whatever is available, but you think that maybe people prefer one brand over others.
So you offer the class a choice of six brands of soft drinks and ask everyone to choose their favorite.
But how are you going to analyze these data?
How can you tell if people truly have a preference for one type of soft drink
or if they are simply choosing one at random?
Most of the statistics that we have used so far
estimate population parameters from a distribution of scores in a sample.
So, for example, the t-test uses sample variance as an estimator of population variance.
Tests like this are called "parametric" tests,
and they work because the central limit theorem shows us that the distribution of sample means
will form an approximately normal distribution.
But there is a class of tests that do not rely upon parameter estimations or
distribution assumptions, and these are called "non-parametric" statistics.
Non-parametrics are especially useful with nominal and ordinal level data, although,
they can be used with interval and ratio level data, as well.
So, for example, you want to make a comparison about whether males or females
are more likely to help a person in either emotional distress or physical peril.
Or perhaps you want to know if patterns of soft drink consumption are the same at a state university as at a private school.
Maybe you want to know if certain dormitories are
over-represented or underrepresented in their number of student government representatives.
Nonparametric statistics have several advantages over parametric statistics.
Number one:
Non-parametric tests are not susceptible to outliers the way that parametric tests are.
One outlier score can affect a parametric test by inflating the variance and hence the error term.
This can invalidate a conclusion drawn by the parametric test. So you make a type I error or perhaps even a type II error.
Outliers in correlation can change the direction or the strength of a relationship between variables,
but non-parametric tests do not have this limitation because they measure central tendency using ranked data
Ranking uses the median rather than a mean.
So, for example, if the tenth number in your distribution is an outlier, that
outlier can really shift the mean when you add it to other numbers.
But when you are considering ranks, it is still just the tenth number no matter how big the actual value is.
So the main advantage of nonparametric statistics is that they do not require
restrictive assumptions about the distribution.
For instance, they do not assume that soft drink consumption is normally distributed.
Non-parametric tests can also be used when you have non-normal, interval, or ratio data, such as data that are highly skewed.
Typically, therefore, non-parametric statistics are only used when the data are too skewed to use a parametric test.
Another advantage is that non-parametric statistics are typically easier to calculate by hand, and we will practice
doing some of them by hand in order to learn them.
However, the main disadvantage of non-parametric statistics
is that they are not as powerful as parametric tests.
That means that they are more likely to miss an effect that truly exists or
that you make a Type II error.
So as you get better with statistics you will be using mostly parametric statistics,
But you should keep some tools in your back pocket that you can use at times that the parametric statistics fail.
Each parametric test has a corresponding
non-parametric test.
Here you can see each parametric test and it's corresponding non-parametric
alternative. So, for example, if you plan to do an independent samples t-test,
but the data were highly skewed you could use a Mann-Whitney U Test,
instead. If you wanted to do a one-way ANOVA,
but instead of groups you were looking for differences over five years of a program,
you could do the Kruskal-Wallis test, instead.
Each of these non-parametric statistics can be conducted in SPSS,
but there is another test called the
"chi-square" test that is an ever-present help in times of statistical trouble, and
It would be the perfect tool for analyzing our soft-drink data.
Anytime you want to know "does my sample look like the population,"
you can use the chi-square.
This adaptability makes chi-square one of the most popular and versatile of the non-parametric statistics.
The Bear Handout on the page about choosing the right statistical test
also has a comparison of the parametric statistics alongside their non-parametric alternatives.